8.4 Basic Geometric Transformations & Trigonometric Ratios

8.4 Basic Geometric Transformations & Trigonometric Ratios

The final section of Chapter 8 bridges spatial visual reasoning with coordinate geometry and basic right-triangle trigonometry. Questions in this domain test your ability to manipulate geometric figures algebraically on the Cartesian coordinate plane and solve real-world distance, height, and angle problems using primary trigonometric ratios.

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Geometric Transformations in the Coordinate Plane

A transformation is an operation that alters the position, orientation, or size of a geometric figure (pre-image) to produce a new figure (image).

Rigid Transformations (Isometries)

Rigid transformations preserve shape and size (congruence). The original figure and transformed figure are identical in dimensions.

1. Translation (Slide): Shifts every point by a fixed distance vector $(a, b)$. $(x, y) \to (x + a, y + b)$
2. Reflection (Flip): Flips a figure across a line of reflection.
   • Across x-axis: $(x, y) \to (x, -y)$
   • Across y-axis: $(x, y) \to (-x, y)$
   • Across line $y = x$: $(x, y) \to (y, x)$
   • Across Origin (0,0): $(x, y) \to (-x, -y)$
3. Rotation (Turn): Rotates a figure around a center point (usually origin $(0,0)$) counterclockwise:
   • 90° CCW Rotation: $(x, y) \to (-y, x)$
   • 180° CCW Rotation: $(x, y) \to (-x, -y)$
   • 270° CCW Rotation (or 90° CW): $(x, y) \to (y, -x)$

Non-Rigid Transformations (Similarity)

1. Dilation (Resize): Resizes a figure using a scale factor $k$ relative to a center point. $(x, y) \to (kx, ky)$
   • If $k > 1$, the transformation is an enlargement.
   • If $0 < k < 1$, the transformation is a reduction.
   • Dilations preserve angles and shape (producing similar figures), but alter side lengths and perimeter by $k$ and area by $k^2$.

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Coordinate Geometry: Distance, Midpoint & Slope

Analytic geometry allows geometric figures to be analyzed using algebraic formulas on the $(x, y)$ coordinate plane.

1. The Distance Formula

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is derived directly from the Pythagorean theorem ($a^2 + b^2 = c^2$):

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Example: To find distance between $(1, 2)$ and $(4, 6)$: $d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

2. The Midpoint Formula

The midpoint $M$ of a line segment connecting $(x_1, y_1)$ and $(x_2, y_2)$ is the average of the coordinates:

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

3. Slope of a Line ($m$)

Slope represents the rate of change or steepness of a line:

$m = \frac{\text{Rise}}{\text{Run}} = \frac{y_2 - y_1}{x_2 - x_1}$

• Parallel Lines: Have identical slopes ($m_1 = m_2$).
• Perpendicular Lines: Have negative reciprocal slopes ($m_1 \cdot m_2 = -1$). For example, if line 1 has slope $\frac{2}{3}$, a perpendicular line has slope $-\frac{3}{2}$.

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Basic Right Triangle Trigonometry (SOH CAH TOA)

Trigonometry is the study of relationships between side lengths and angles in triangles. For AFQT Mathematics Knowledge, you only need to master the primary trigonometric ratios for right-angled triangles.

                  Right Triangle Ratios
                  
                         /|
                        / |
          Hypotenuse   /  |  Opposite (to θ)
             (c)      /   |
                     / θ  |
                    +-----+ 
                   Adjacent (to θ)

The SOH CAH TOA Mnemonic

$$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \text{(SOH)}$$ $$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \text{(CAH)}$$ $$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \quad \text{(TOA)}$$

Fundamental Trigonometric Identities

• Tangent Ratio Identity: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$
• Pythagorean Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$
• Complementary Angle Rule: $\sin(\theta) = \cos(90^\circ - \theta)$

Essential Trigonometric Values for Benchmark Angles

AFQT questions do not allow calculators. Therefore, trig questions involve benchmark angles (30°, 45°, 60°) derived from special right triangles:

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Angles of Elevation & Depression

Word problems frequently test spatial applications using angles of elevation and depression:

• Angle of Elevation: The angle measured upward from a horizontal reference line to an object above.
• Angle of Depression: The angle measured downward from a horizontal reference line to an object below.

$\text{Horizontal Reference Line} \quad \Longleftrightarrow \quad \text{Angle of Elevation} = \text{Angle of Depression}$

Because horizontal lines are parallel, the angle of elevation from observer A to object B is always equal to the angle of depression from object B to observer A (alternate interior angles).

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AFQT Step-by-Step Problem Walkthroughs

Example 1: Coordinate Transformation Sequence

Problem: Point $P$ has coordinates $(-3, 5)$. Point $P$ is reflected across the x-axis, and then translated by vector $(4, -2)$. What are the final coordinates of $P''$?

1. Step 1 - Reflection across x-axis: $(x, y) \to (x, -y)$. $P(-3, 5) \to P'(-3, -5)$
2. Step 2 - Translation: $(x + 4, y - 2)$. $P'(-3, -5) \to P''(-3 + 4, -5 - 2) = P''(1, -7)$
3. Final Coordinates: $(1, -7)$.

Example 2: Trig Height Word Problem

Problem: A searchlight operator spots an object directly above a tower. An observer standing 100 feet away from the base of the tower measures an angle of elevation of 60° to the top of the tower. How tall is the tower?

1. Identify components: Angle $\theta = 60^\circ$, Adjacent side $= 100$ feet, Opposite side $= h$ (height).
2. Select trig ratio: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$.
3. Set up equation: $\tan(60^\circ) = \frac{h}{100}$.
4. Substitute value: $\sqrt{3} = \frac{h}{100} \implies h = 100\sqrt{3}$ feet.
5. Decimal approximation: $100 \times 1.732 = 173.2$ feet.

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Key Takeaways Summary

• Reflections: x-axis $(x, -y)$, y-axis $(-x, y)$, line $y=x \to (y, x)$.
• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Perpendicular lines have negative reciprocal slopes ($m_1 m_2 = -1$).
• SOH CAH TOA: $\sin = \frac{\text{Opp}}{\text{Hyp}}$, $\cos = \frac{\text{Adj}}{\text{Hyp}}$, $\tan = \frac{\text{Opp}}{\text{Adj}}$.
• Benchmark Trig Values: $\sin(30^\circ)=0.5$, $\tan(45^\circ)=1$, $\sin(60^\circ)=\frac{\sqrt{3}}{2}$.